Rough. ) assumptions, proof. Det Let later. statements. string. statements. ( the. di ) by proof. assumption. Oli. logical. Defied. f L. Eton.

Transcription

1 CHPTR Deductions

2 Oli Qi there be formulas formulas Qi i axioms Deductions What do mean by proof? Rough dea : Using some axioms assumptions continue to infer new true statements until arrive at what wanted to prove So statements of true will be a a proof string each statement should either be ± ~ SF an axiom inferred e g P assumption or statements from earlier P 7 Q or want extra properties Q f like decidability Det Let later be a set at L a { be a set of L ( lo o D be a finite sequence Defined fixed & ( the logical axioms ( the non log Qu of L formulas it Then call D a deduction from if for all Kien of either D Defied 6 of u R or f L is a rule of inference ( T di / later fixed * ton say with M { to Disa deduction from of on write

3 form bout N Cr Ol among other things will want every L is valid ie FL RS preserve truth ie T t do = Let a Let rules of inference be { ( B 0 one L } Let Fx Pcx x Plum { Pcv u u Pcu Pcv Pu p ( u ul Pcu a a 1 Show t Plain Pcu Pcu u Pcu u htxpcx x Roti P ( v u Plum Pcu P( u Plum Rof u u v v ncorrect bed das pone us xplain why tf Pcu u Det ssuming N define Thing the R are fixed = Lol to } n last Thing = example u { Pcu u [ all formulas deducible from Pcu u the theorems generated by via deductions

4 if u M f Prop L 4 Thing is the smallest set C at such formulas that C C rt ( T Q is a 1 Show Thing Rat with satisfies o f L UN ( x is a deduction of L So d Thing ofc Thus u Thun TC Rot with The ( Milo be a Rembert Thus D u show is finite So M = ( x there is a deduction Di dm } Thing of di from Dm u { of } ( with obvious meaning is a deduction of from Let C Thing length that if C satisfies Proceed by at the shortest deduction them induction Thr at on [ lo Then C the from ( M Otherwise ul C Of C may assume there is a 0 set TC ( induction by Thus Rof by Of C C D

5 formula N x yn i term Logical xioms These will be about equality quantifiers Dent The set of logical axioms is defined as follows : For every variable x in L fly x is in ( l x For Xu X all variables i Yi Yu all function symbols f Xn 7 f C x ( [ (X =y r ( xn ] = C Xu Yu ][RCx ( [ C =y all relation symbols R xn Rly Yu Ya are in For every every L for x in Of variable in x L every L t of if t is substitutable ( Qi Ctx Col 4 ( Q 01 G x Lol There are in N are no other formulas in Q : how many formulas are in N?

6 Propositional g v 4 Rules at nference Consequence Remember logic from 108? had things like Here n (V ( n B ( Demons equivalent logically re propositional B re always assigned could Prove things with a truth table variables values of Tor F like the one above Then B VB TC VB 7 71 C CB F F F F F T F F T T F F T T T lso Same! a propositional formula is a tautology if it is always true 7 UR e Ga always true! now define a function to convert first order formulas to propositi at formulas

7 variable ( Running xample Let P := x ( ind all sub formulas at at the form B that are Not in the scope systematically replace prop of another quantifier them with a Repeat for all such sub formulas Systematically replace all remaining atomic formulas with new prop variables Bp CN B V B Notice that Bp is a tautology since BC t B Bp T T T T it T FT * ± TF Fact Bp a tautology pp is valid ie FB converse is not ( always

8 formulas form of using Lemma 4 of Let T = Det d du } be a finite set of uniformly L formulas Of another L na = Tp dip Lnp Op be the results at applying the above conversion procedure Let to all at the L is a proposition say consequence that of ^ T if See x 44 [ Lip p Lnp ] Gp is a tautology The Rules of inference Whenever is a propositional cons T ( Pc ( T is a rule of inference For all formulas Of 4 variables x that are Not all free in 4 ( QR ( 1474 ( e4y # y one rules of inference There are no other rules

9 from from form To clarify o f you prone L B [ are 7017 p is a tautology you can conclude Of Provided x is Not free in 4 4 Of y of you u can deduce 4 Vx " 474 mportant Note o The set of Rof s is decidable : there is algorithm ( think coup that given M L prog a finite set at L formulas can decide C in a finite amount at time if CT d is a not Rat or lso is decide able

10 an f ( 5 Soundness The goal of this section is to prone : Theorem 5 Soundness f to to [ of our deductive system Roti * is a set at formulas Soundness says X if assuming ( structure that thinks can prove Y any is true will also think of is true OR if can prove it OR it's true " " " proofs preserve " truth Recap of requirements for our deductive exercises { system : decide M t 0 if 7 on not an algorithm can algorithm can decide finite T given if ( T Q Rot is a Rot M is finite for ( T o every 4 : 5 Rof s truth if ( M o is preserve an Rof

11 S for n Theorem 5 t Rt Let Let s be any Uaf show M t ] Note : is of type l L Q1 or Q Claim : f L has type M Cs ] pt of claim Kd o L has the form ( X Lcs y n a yn ( Rex xd 7 Rly ynd o LN LST MLS] ssure M S C x M ( said Cy n saw e RM Thus M Mf ( San Claim : f L has type S C xn = SC yn scyul R o L has the form x d Q where t is sub X Q ssure Me ( tx d [ s ] so M cfsfcxlm for all MCM MFcf@Cx5LtDJoNeedThn6Z Thus : M 0/( Mff :C s ] so M t o Thus M Of Cs ] so M Lcs

12 5 0 will show M to [ rcxlm ] ] ssure Claim f L has type l Z Q Then Mf LST pls at claim : book + exercises D Then f ( T Q is a Rat T to Pf Let DM be an structure M TST for every Va f S ; must show vaf M t ] for every r Otr Let r be an arbitrary Claim : f CT PC Mf 0 C o has type t see ore book M 0 Claim : f ( QR them n has type rt ( r 0 has the form 0 1 ( ly 4 yo or ( to 74 x4 y only not free in 4 with x Cr] So lso Uaf s treat 1 ; is an exercise assure M f c s eggb assure Mt 4 Cr ] : * for WTS M 4 Cr ] To show Mf Vx Offer ] let me Mj

13 if Let ( 5 Observe M yer ] by M y [ RL xlm ] ] BB since x is net free in y ( Prop 17 7 lso M 4 [ rcxlmd or Mfcfcrcxlm ] ] by B ( with s = r ex ms Thus M Of [ rcxlm ] ] 1 Theorem 5 pet # Soundness f to formulas Let thing formulas provable = = { to let X ; implied logically by w TS that if Thing = f X j use Prop 4 to from 4 t 4 which says X z X! ( T o is a Rat TX with o c X they Thing Let 's check the list X MK Mt Certainly Since f ( l so it's X certainly true that M M so X ( P 0 be an Raf with TX ssure M ; WTS M MX so M t M =P Mf M so by Then 5 t Now O Thus M f Q y

14 relation of system of Let Note diff thanx 7 Properties of our deductive system him 7 Our deal can prone that = is an equiv That is 1 t x x t x y y=x t ( x = yay z x = Z pt see book Lem 7 to iff t Vx Q pet s sue t 0 y be a war Deduction of x Q ( insert deal y y = y Q 0 Oy pc : a ( y y to yay VX 0 QR : not free in Y Y it XO pc : ( y=y My= yeux OH Vx 0 ( ssure t x 0 0 is tie sore as 0 Deduction at Q ( insert deal Fx 0 Fx Q tx 0 Q Q1 G Pc D

15 1 Suppose y f 7 Lemma 7 Suppose to Let L a xp ' is with L replaced by p ' to Let y :=f L f ' is with L rep! by 8 ' to pet By the previous 't lemma so as t 0 't G The 74 ( Deduction Then Let Q be a sentence a set of formulas Then for any formula Of vo to iff of rt ( ssure t ( o Then as u 0 to vo t Of ( by PC sse vo to Thur show quo Let X y t lo 41 that of C X X implying 4 which is what want We use Prop { ± o X by b X by O o 7 X by PC Pcr PC : is true true o is a tautology Let 47 bear Rosi with Let M K } T X So t Gri

16 at o type : PC L is a prop = a so is a " cons " 8 " { o 8 Vu } 0 K ] =P ' t since of a t r ' o = L X QR ( universal type 17=105 L := p Fxs with x not free in B rex to p s t Corp or by Pc = t Grp Vx of G is a sent so not free in 0 = to ( ex or p LC X or in by assumes type QR C ext : similar Thus Prop 4 applies y